A solid sphere of mass ' \(m\) ' and radius ' \(r\) ' is allowed to roll without slipping from the highest point of an inclined plane of length ' \(L\) ' and makes an angle \(30^{\circ}\) with the horizontal. The speed of the particle at the bottom of the plane is \(v_1\). If the angle of inclination is increased to \(45^{\circ}\) while keeping \(L\) constant. Then the new speed of the sphere at the bottom of the plane is \(v_2\). The ratio \(v_1^2: v_2^2\) is:

A block of mass \(10\,kg\) is moving along \(x\)-axis under the action of force \(F=5 x\; N\). The work done by the force in moving the block from \(x =2\,m\) to \(4\,m\) will be ............\(J\).

One mole of an ideal gas passes through a process where pressure and volume obey the relation \(P\, = {P_0}\,\left[ {1 - \frac{1}{2}{{\left( {\frac{{{V_0}}}{V}} \right)}^2}} \right]\).  Here \(P_0\) and \(V_0\) are constants. Calculate the change in the temperature of the gas if its volume change from \(V_0\) to \(2V_0\)

The distance between object and its \(3\) times magnified virtual image as produced by a convex lens is \(20 \mathrm{~cm}\). The focal length of the lens used is _______ \(\mathrm{cm}\).

Consider a sample of oxygen behaving like an ideal gas. At \(300 \,K ,\) the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be : (Molecular weight of oxygen is \(32 \,g / mol\) \(\left. R =8.3 \,J K ^{-1} mol ^{-1}\right)\)

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

In an expression a \(\times 10^{\mathrm{b}}\) _______.

A thermally insulted vessel contains \(150\, g\) of water at \(0\,^oC\). Then the air from the vessel is pumped out a adiabatically. A fraction of water turns into ice and the rest evaporates at \(0\,^oC\) itself. The mass of evaporated water will be closes to ....... \(g\) (Latent heat of vaporization of water \(= 2.10 \times10^6\, Jkg^{-1}\) and Laten heat of Fusion of water \( = 3.36 \times10^5\,Jkg^{-1}\) )

A plane electromagnetic wave is moving in free space with velocity \(c =3 \times 10^8 m / s\) and its electric field is given as \(\overrightarrow{ E }=54 \sin ( kz -\omega t ) \hat{ j } V / m\), where \(\hat{j}\) is the unit vector along \(y\)-axis. The magnetic field vector \(\overrightarrow{ B }\) of the wave is :

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Emission of electrons in photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photoemissive substance.
Reason (R) : A negative electric potential, which stops the emission of electrons from the surface of a photoemissive substance, varies linearly with frequency of incident radiation.
In the light of the above statements, choose the most appropriate answer from the options given below :

For a nucleus \({ }_Z^A X\) having mass number \(A\) and atomic number \(Z\) \(A.\) The surface energy per nucleon \(\left(b_s\right)=a_1 A^{2 / 3}\) \(B.\) The Coulomb contribution to the binding energy \(b_c=-a_2 \frac{Z(Z-1)}{A^{4 / 3}}\) \(C.\) The volume energy \(b_v=a_3 A\) \(D.\) Decrease in the binding energy is proportional to surface area. \(E.\) While estimating the surface energy, it is assumed that each nucleon interacts with \(12\) nucleons, \(\left(a_1, a_2\right.\) and \(a _3\) are constants) Choose the most appropriate answer from the options given below:

In a nuclear fission reaction of an isotope of mass \(M\), three similar daughter nucler of same mass are formed. The speed of a daughter nuclei in terms of mass defect \(\Delta \mathrm{M}\) will be _______.

If \(\mathrm{S}=\left\{\mathrm{z} \in \mathbb{C}: \frac{\mathrm{z}-i}{\mathrm{z}+2 i} \in \mathbb{R}\right\}\), then :